How to Read Two Sided Stem Plot

Stalk and leafage plots

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• Elements of a good stalk and leafage plot
• Tips on how to draw a stem and leaf plot
  • Instance ane – Making a stalk and leaf plot
• The master reward of a stalk and leaf plot
  • Example 2 – Making a stem and foliage plot
  • Example 3 – Making an ordered stem and leaf plot
• Splitting the stems
  • Example four – Splitting the stems
  • Case 5 – Splitting stems using decimal values
• Outliers
• Features of distributions
• Using stalk and leaf plots every bit graphs
  • Example 6 – Using stem and leaf plots every bit graph

A stalk and leaf plot, or stem plot, is a technique used to classify either detached or continuous variables. A stem and leaf plot is used to organize data as they are collected.

A stem and leaf plot looks something like a bar graph. Each number in the information is broken down into a stem and a foliage, thus the name. The stem of the number includes all merely the concluding digit. The foliage of the number will always exist a unmarried digit.

Elements of a good stalk and leaf plot

A good stem and leafage plot

• shows the first digits of the number (thousands, hundreds or tens) equally the stem and shows the last digit (ones) as the leaf.
  
• usually uses whole numbers. Annihilation that has a decimal point is rounded to the nearest whole number. For example, exam results, speeds, heights, weights, etc.
  
• looks like a bar graph when information technology is turned on its side.
  
• shows how the data are spread—that is, highest number, everyman number, most mutual number and outliers (a number that lies outside the master group of numbers).

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Tips on how to draw a stalk and foliage plot

Once y'all have decided that a stem and leaf plot is the best style to show your data, draw it equally follows:

• On the left manus side of the folio, write downwards the thousands, hundreds or tens (all digits merely the final one). These will be your stems.
• Draw a line to the right of these stems.
• On the other side of the line, write downwards the ones (the last digit of a number). These will be your leaves.

For example, if the observed value is 25, then the stalk is 2 and the leaf is the v. If the observed value is 369, then the stem is 36 and the leaf is 9. Where observations are authentic to ane or more decimal places, such as 23.seven, the stem is 23 and the leaf is 7. If the range of values is too keen, the number 23.7 tin be rounded upwards to 24 to limit the number of stems.

In stalk and leaf plots, tally marks are non required because the actual data are used.

Not quite getting it? Try some exercises.

Example 1 – Making a stem and leaf plot

Each morning, a teacher quizzed his form with 20 geography questions. The course marked them together and everyone kept a record of their personal scores. As the year passed, each student tried to improve his or her quiz marks. Every twenty-four hour period, Elliot recorded his quiz marks on a stalk and foliage plot. This is what his marks looked like plotted out:

Tabular array ane. Elliot'south scores on the bones facts quiz concluding year

Stem	Leaf
0	3 half-dozen 5
1	0 one 4 three 5 6 5 six eight 9 7 nine
ii	0 0 0 0

Analyse Elliot'due south stem and foliage plot. What is his most mutual score on the geography quizzes? What is his highest score? His everyman score? Rotate the stem and leaf plot onto its side so that information technology looks like a bar graph. Are most of Elliot's scores in the 10s, 20s or under 10? It is hard to know from the plot whether Elliot has improved or not because we exercise not know the order of those scores.

Try making your own stem and foliage plot. Use the marks from something like all of your exam results final yr or the points your sports squad accumulated this season.

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The main advantage of a stem and leaf plot

The primary advantage of a stem and leaf plot is that the information are grouped and all the original data are shown, too. In Example iii on battery life in the Frequency distribution tables section, the tabular array shows that two observations occurred in the interval from 360 to 369 minutes. Still, the table does not tell you what those bodily observations are. A stem and leafage plot would show that information. Without a stalk and leaf plot, the 2 values (363 and 369) can only be found by searching through all the original data—a tedious task when you have lots of data!

When looking at a data set, each observation may be considered as consisting of two parts—a stem and a leaf. To make a stem and leafage plot, each observed value must first be separated into its ii parts:

• The stalk is the kickoff digit or digits;
• The leaf is the final digit of a value;
• Each stem can consist of any number of digits; only
• Each leafage can accept just a single digit.

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Example 2 – Making a stem and leaf plot

A teacher asked ten of her students how many books they had read in the last 12 months. Their answers were equally follows:

12, 23, 19, 6, ten, 7, 15, 25, 21, 12

Prepare a stem and leaf plot for these data.

Tip: The number 6 can be written as 06, which means that it has a stalk of 0 and a leaf of half-dozen.

The stalk and leaf plot should await similar this:

Tabular array ii. Books read in a year by x students

Stem	Foliage
0	6 7
1	2 ix 0 5 2
2	3 5 one

In Table 2:

• stem 0 represents the class interval 0 to ix;
• stem 1 represents the class interval 10 to 19; and
• stem two represents the course interval 20 to 29.

Usually, a stem and leafage plot is ordered, which merely means that the leaves are bundled in ascending order from left to correct. Also, at that place is no need to separate the leaves (digits) with punctuation marks (commas or periods) since each leaf is ever a unmarried digit.

Using the data from Table ii, we fabricated the ordered stem and leafage plot shown below:

Tabular array 3. Books read in a yr by 10 students

Stalk	Leafage
0	six 7
ane	0 2 2 5 nine
2	1 three 5

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Example iii – Making an ordered stem and leaf plot

Fifteen people were asked how often they collection to work over 10 working days. The number of times each person drove was as follows:

5, seven, nine, ix, three, five, 1, 0, 0, 4, 3, seven, 2, 9, eight

Brand an ordered stem and leaf plot for this table.

It should be fatigued as follows:

Table 4. Number of drives to work in ten days

Stem	Foliage
0	0 0 ane two 3 3 4 5 v 7 7 8 ix 9 ix

Splitting the stems

The organization of this stem and leaf plot does non give much information nigh the data. With simply one stem, the leaves are overcrowded. If the leaves go too crowded, so it might exist useful to dissever each stem into two or more components. Thus, an interval 0–9 can exist split into 2 intervals of 0–four and 5–9. Similarly, a 0–9 stem could be divide into 5 intervals: 0–one, 2–3, 4–v, six–vii and eight–9.

The stalk and leaf plot should then wait similar this:

Tabular array 5. Number of drives to piece of work in x days

Stem	Foliage
0(0)	0 0 1 2 iii 3 4
0(5)	5 five 7 vii 8 9 9 ix

Note: The stem 0⁽⁰⁾ ways all the data within the interval 0–4. The stalk 0⁽⁵⁾ ways all the data within the interval v–ix.

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Example 4 – Splitting the stems

Britney is a swimmer preparation for a competition. The number of 50-metre laps she swam each day for thirty days are every bit follows:

22, 21, 24, 19, 27, 28, 24, 25, 29, 28, 26, 31, 28, 27, 22, 39, twenty, 10, 26, 24, 27, 28, 26, 28, xviii, 32, 29, 25, 31, 27

1. Ready an ordered stem and foliage plot. Make a brief comment on what information technology shows.
2. Redraw the stalk and leaf plot by splitting the stems into five-unit intervals. Make a brief annotate on what the new plot shows.

Answers

1. The observations range in value from x to 39, so the stem and leaf plot should take stems of 1, two and 3. The ordered stem and leaf plot is shown below:

   Table 6. Laps swum by Britney in xxx days

   Stalk	Foliage
   1	0 8 nine
   2	0 one two 2 4 4 iv 5 5 6 6 6 7 7 7 7 viii 8 viii 8 8 9 9
   three	1 i ii 9

   The stalk and foliage plot shows that Britney usually swims betwixt 20 and 29 laps in training each solar day.
2. Splitting the stems into v-unit of measurement intervals gives the following stalk and leafage plot:

   Tabular array vii. Laps swum by Britney in 30 days

   Stem	Leafage
   1(0)	0
   1(v)	eight 9
   2(0)	0 i 2 2 4 four 4
   2(5)	five v half-dozen 6 6 seven vii vii 7 8 viii 8 8 viii 9 9
   3(0)	ane 1 two
   iii(five)	ix

   Note: The stem ane⁽⁰⁾ means all data betwixt 10 and fourteen, one^(five) means all data between xv and 19, and and then on.

   The revised stem and leafage plot shows that Britney ordinarily swims between 25 and 29 laps in training each mean solar day. The values 1⁽⁰⁾ 0 = 10 and three⁽⁵⁾ ix = 39 could exist considered outliers—a concept that volition be described in the next section.

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Case five – Splitting stems using decimal values

The weights (to the nearest tenth of a kilogram) of thirty students were measured and recorded as follows:

59.2, 61.5, 62.3, 61.4, 60.9, 59.8, 60.5, 59.0, 61.1, 60.vii, 61.6, 56.3, 61.ix, 65.7, 60.four, 58.9, 59.0, 61.2, 62.1, 61.iv, 58.4, lx.8, 60.2, 62.7, 60.0, 59.3, 61.ix, 61.7, 58.four, 62.ii

Set an ordered stem and leaf plot for the data. Briefly comment on what the analysis shows.

Answer

In this case, the stems will exist the whole number values and the leaves will be the decimal values. The data range from 56.3 to 65.seven, so the stems should start at 56 and finish at 65.

Table 8. Weights of 30 students

Stem	Leaf
56	3
57
58	4 4 9
59	0 0 ii three eight
sixty	0 2 4 5 7 viii nine
61	one 2 4 4 5 vi 7 ix ix
62	1 2 three 7
63
64
65	7

In this instance, it was not necessary to carve up stems because the leaves are non crowded on too few stems; nor was information technology necessary to circular the values, since the range of values is not large. This stem and foliage plot reveals that the group with the highest number of observations recorded is the 61.0 to 61.9 grouping.

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Outliers

An outlier is an extreme value of the data. It is an ascertainment value that is significantly different from the residual of the data. There may be more than i outlier in a prepare of data.

Sometimes, outliers are pregnant pieces of information and should not be ignored. Other times, they occur because of an fault or misinformation and should be ignored.

In the previous example, 56.iii and 65.7 could be considered outliers, since these two values are quite dissimilar from the other values.

Past ignoring these two outliers, the previous example'due south stem and leaf plot could be redrawn as beneath:

Tabular array ix. Weights of 30 students except for outliers

Stem	Leaf
58	4 4 9
59	0 0 2 3 viii
60	0 ii 4 five vii 8 nine
61	one 2 4 four 5 6 seven ix 9
62	ane two 3 7

When using a stalk and leaf plot, spotting an outlier is ofttimes a matter of judgment. This is because, except when using box plots (explained in the department on box and whisker plots), there is no strict rule on how far removed a value must be from the rest of a data set to qualify as an outlier.

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Features of distributions

When yous assess the overall pattern of any distribution (which is the pattern formed by all values of a detail variable), look for these features:

• number of peaks
• full general shape (skewed or symmetric)
• centre
• spread

Number of peaks

Line graphs are useful because they readily reveal some feature of the data. (Come across the section on line graphs for details on this blazon of graph.)

The starting time characteristic that tin can exist readily seen from a line graph is the number of high points or peaks the distribution has.

While about distributions that occur in statistical data accept only one main peak (unimodal), other distributions may have ii peaks (bimodal) or more than than two peaks (multimodal).

Examples of unimodal, bimodal and multimodal line graphs are shown beneath:

General shape

The second principal feature of a distribution is the extent to which it is symmetric.

A perfectly symmetric curve is one in which both sides of the distribution would exactly match the other if the effigy were folded over its primal betoken. An example is shown below:

A symmetric, unimodal, bell-shaped distribution—a relatively common occurrence—is called a normal distribution.

If the distribution is lop-sided, it is said to be skewed.

A distribution is said to be skewed to the right, or positively skewed, when virtually of the data are full-bodied on the left of the distribution. Distributions with positive skews are more common than distributions with negative skews.

Income provides one instance of a positively skewed distribution. About people make under $forty,000 a yr, but some make quite a bit more than, with a smaller number making many millions of dollars a year. Therefore, the positive (right) tail on the line graph for income extends out quite a long way, whereas the negative (left) skew tail stops at zero. The correct tail clearly extends farther from the distribution's center than the left tail, every bit shown below:

A distribution is said to exist skewed to the left, or negatively skewed, if nigh of the data are concentrated on the correct of the distribution. The left tail clearly extends farther from the distribution's center than the right tail, as shown below:

Center and spread

Locating the center (median) of a distribution tin can exist done by counting half the observations up from the smallest. Obviously, this method is impracticable for very large sets of data. A stem and foliage plot makes this easy, yet, because the information are arranged in ascending society. The mean is another mensurate of central tendency. (See the chapter on central tendency for more detail.)

The corporeality of distribution spread and any large deviations from the general pattern (outliers) can be quickly spotted on a graph.

Using stem and leaf plots equally graphs

A stem and leafage plot is a simple kind of graph that is fabricated out of the numbers themselves. Information technology is a means of displaying the main features of a distribution. If a stem and leaf plot is turned on its side, information technology will resemble a bar graph or histogram and provide similar visual data.

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Example 6 – Using stalk and leaf plots every bit graph

The results of 41 students' math tests (with a all-time possible score of seventy) are recorded below:

31, 49, xix, 62, 50, 24, 45, 23, 51, 32, 48, 55, 60, twoscore, 35, 54, 26, 57, 37, 43, 65, fifty, 55, 18, 53, 41, 50, 34, 67, 56, 44, 4, 54, 57, 39, 52, 45, 35, 51, 63, 42

1. Is the variable discrete or continuous? Explicate.
2. Gear up an ordered stem and leaf plot for the data and briefly draw what information technology shows.
3. Are there whatsoever outliers? If then, which scores?
4. Look at the stem and leafage plot from the side. Describe the distribution'due south main features such as:
   1. number of peaks
   2. symmetry
   3. value at the centre of the distribution

Answers

1. A test score is a discrete variable. For example, information technology is not possible to accept a exam score of 35.74542341....
2. The lowest value is 4 and the highest is 67. Therefore, the stalk and leaf plot that covers this range of values looks like this:
   

   Tabular array ten. Math scores of 41 students

   Stem	Leafage
   0	four
   1	8 nine
   2	iii four 6
   iii	one two 4 five 5 vii 9
   four	0 1 two 3 iv 5 5 eight ix
   5	0 0 0 one 1 ii 3 4 4 five 5 vi 7 7
   6	0 2 3 5 7

   Note: The annotation 2|four represents stem 2 and foliage 4.

   The stalk and leaf plot reveals that nearly students scored in the interval between 50 and 59. The large number of students who obtained high results could hateful that the test was too piece of cake, that well-nigh students knew the cloth well, or a combination of both.

3. The event of 4 could be an outlier, since at that place is a large gap between this and the side by side effect, xviii.
4. If the stalk and leaf plot is turned on its side, it volition look similar the post-obit:

   

   The distribution has a single summit inside the 50–59 interval.

   Although at that place are only 41 observations, the distribution shows that most data are clustered at the right. The left tail extends farther from the data centre than the right tail. Therefore, the distribution is skewed to the left or negatively skewed.

   Since there are 41 observations, the distribution centre (the median value) will occur at the 21st observation. Counting 21 observations up from the smallest, the centre is 48. (Note that the same value would have been obtained if 21 observations were counted down from the highest observation.)

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Source: https://www150.statcan.gc.ca/n1/edu/power-pouvoir/ch8/5214816-eng.htm

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